Algebra

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Algebra (from the Arabic "al-jabr" , literally "broken part reunion") is one of the broadest parts of mathematics, along with number theory, geometry and analysis. In its most common form, algebra is the study of mathematical symbols and rules for manipulating these symbols; it is a unifying thread of almost any math. Thus, it covers everything from the completion of basic equations to abstraction studies such as groups, rings, and fields. A more basic algebraic base is called basic algebra; the more abstract part is called abstract algebra or modern algebra. Basic algebra is generally considered essential for any study of mathematics, science, or engineering, as well as applications such as medicine and economics. Abstract algebra is a major field in advanced mathematics, studied mainly by professional mathematicians.

Basic algebra is different from arithmetic in the use of abstraction, such as using letters to stand for numbers that are either unknown or allowed to take many values. For example, in ${\ displaystyle x 2 = 5}$ ${\ displaystyle x}$ is unknown, but inverse law can be used to find its value: ${\ displaystyle x = 3}$ . In mc ² , the letter ${\ displaystyle E}$ and ${\ displaystyle m}$ is a variable, and the letters ${\ displaystyle c}$ is the constant, the speed of light in a vacuum. Algebra provides a method for writing formulas and solving equations that are much clearer and easier than the old method of writing them all in words.

The word algebra is also used in a particular way. The type of special mathematical object in abstract algebra is called "algebra", and the word is used, for example, in linear algebraic phrases and algebraic topologies.

A mathematician who conducts research in algebra is referred to as algebra .

Video Algebra

Etimologi

The word algebra comes from the Arabic language ????? ( al-jabr illuminates "damaged reunion") from the title of the book Ilm al-jabr wa'l-mu ?? reinforcements by the mathematicians and astronomers of Persia al-Khwarizmi. The word entered English during the 15th century, from Spanish, Italian, or Latin of the Middle Ages. This initially refers to a surgical procedure of setting a broken bone or a dislocation. The meaning of mathematics was first recorded in the sixteenth century.

Maps Algebra

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as one word or with qualification.

• As one word without articles, "algebra" gives the name of a broad section of mathematics.
• As a single word with an article or in plural, "algebra" or "algebra" denotes certain mathematical structures, whose exact definition depends on the author. Typically, the structure has the addition, multiplication, and scalar multiplication (see Algebra above the field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to the generalization of the above concept, which allows for n-ary operation.
• With qualifications, there is the same difference:
  • Without the article, it means part of algebra, such as linear algebra, basic algebra (rules of manipulation-symbols taught in the elementary class of mathematics as part of primary and secondary education), or abstract algebra (studying algebraic structures for self themselves).
  • With an article, it means an instance of some abstract structure, such as Lie algebra, association algebra, or vertex operator algebra.
  • Sometimes both meanings exist for the same qualifications, as in the sentence: commutative algebra is the study of commutative rings, which are commutative algebra over integers .

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Algebra as a branch of mathematics

Aljabar dimulai dengan perhitungan yang mirip dengan aritmatika, dengan huruf yang berdiri untuk angka. Ini memungkinkan bukti properti yang benar tidak peduli nomor yang terlibat. Misalnya, dalam persamaan kuadrat

$            {\displaystyle         a        x^{            2          }                b        x                c        =        0        ,      }        \{\backslash \; displaystyle\; axe\; ^\; \{2\}\; bx\; c\; =\; 0,\}\;  ÂÂ$

${\ displaystyle a, b, c}  ÂÂ$ dapat berupa angka apa pun (kecuali ${\ displaystyle a}  ÂÂ$ tidak dapat ${\ displaystyle 0}  ÂÂ$ ), dan rumus kuadrat dapat digunakan untuk menemukan nilai dengan cepat dan mudah dari kuantitas yang tidak diketahui ${\ displaystyle x}  ÂÂ$ yang memenuhi persamaan. Artinya, untuk menemukan semua solusi dari persamaan.

Historically, and in today's teaching, algebra studies begin with the solution of equations like the above quadratic equation. Then the more general questions, such as "does the equation have solutions?", "How many solutions does the equation have?", "What can be said about the nature of the solution?" considered. These questions lead to the idea of ​​form, structure and symmetry. This development allows expanded algebra to consider non-numerical objects, such as vectors, matrices, and polynomials. The structural properties of these non-numerical objects are then abstracted to define algebraic structures such as groups, rings, and planes.

Before the 16th century, mathematics was divided into just two subfields, arithmetic and geometry. Although several methods, which have been developed long before, can be considered today as algebra, the appearance of algebra and, shortly thereafter, the infinitesimal calculus as a subset of mathematics only dates from the 16th or 17th centuries. From the second half of the 19th century, many new areas of mathematics emerged , mostly using arithmetic and geometry, and almost all use algebra.

Currently, algebra has grown to include many branches of mathematics, as can be seen in Mathematics Subject Classification where no first level area (two-digit entry) is called algebra . Algebra today includes parts of the general 08-algebra system, 12-field theory and polynomials, 13-commutative algebra, 15-linear and multilinear algebra; matrix theory, 16-associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homologous algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

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History

The history of algebra

The algebraic roots can be traced back to ancient Babylon, which developed an advanced arithmetic system by which they were able to perform calculations in an algorithmic way. The Babylonians developed a formula for calculating solutions to problems usually solved today by using linear equations, quadratic equations, and indefinite linear equations. By contrast, most Egyptians of this age, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solve these equations by geometric methods, as described in Rhind Mathematical Papyrus, Euclid > Element , and Nine Chapters on Mathematical Art . The geometrical work of the Greeks, characterized in the Elements , provides a framework for generalizing formulas beyond the solution of certain problems into more general systems for expressing and resolving equations, although this will not be realized until mathematics is developed. in medieval Islam.

At the time of Plato, Greek mathematics had undergone drastic changes. The Greeks created a geometric algebra in which terms are represented by sides of geometric objects, usually lines, which have letters associated with it. Diophantus (3rd century AD) was a Greek mathematician of Alexandria and author of a series of books called Arithmetica . These texts deal with solving algebraic equations, and have led, in number theory to the modern idea of ​​the Diophantine equation.

The previous tradition discussed above has a direct influence on Persian mathematicians, Mu'ammad ibn M? al-Khw? rizm? (c 780-850). He then wrote a Book of Completion of Calculations with Completion and Balancing , which formed algebra as a mathematical discipline that did not depend on geometry and arithmetic.

The Hellenistic mathematicians of Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continue Egyptian and Babylonian traditions, although Diophantus' Arithmetica and Brahmagupta's
Hmasphu? Asiddh? Nta is at a higher level. For example, the first complete arithmetic solution (including zero and negative solutions) for quadratic equations is described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Persian and Arab mathematicians developed the algebraic method to a much higher degree of sophistication. Although Diophantus and Babylonia mostly use ad hoc special methods to solve equations, Al-Khawarizmi's contribution is very basic. He solves linear and quadratic equations without algebraic symbolism, negative or zero numbers, so he must distinguish several types of equations.

In the context in which algebra is identified with the theory of equations, Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in a context in which he is identified with rules to manipulate and solve equations, the Persian al-Khwarizmi mathematician is considered the "father of algebra". The debate now exists whether what (in the general sense) has more right to be known as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more basic than the algebra found in Arithmetica and Arithmetica is temporary syncopation Al-Jabr is completely rhetorical. Those who support Al-Khawarizmi point to the fact that it introduces the method of "subtraction" and "balancing" (transposition of diminished terms to the other side of an equation, that is, the cancellation of similar terms on the opposite side of the equation) i> al-jabr was originally called, and that it provides a complete explanation for solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right. The algebra also no longer cares "with a series of issues to be solved, but an exposition that begins with a primitive term in which the combination should give all possible prototypes to the equation, which therefore explicitly form the actual object of study". He also studied equations for himself and "in general, to the extent that not only arose in the course of solving problems, but specifically called to define the class of infinite problems".

Another Persian mathematician, Omar Khayyam, is credited with identifying the foundations of algebraic geometry and finding common geometric solutions of cubic equations. His Book of Algebra Problem Demonstration (1070), which laid down the principles of algebra, was part of the Persian mathematical body that was eventually transmitted to Europe. However, another Persian mathematician, Sharaf al-D? N al-T? S?, Invented algebraic and numerical solutions for various cases of cubic equations. He also developed the concept of function. Indian mathematicians Mahavira and Bhaskara II, Persian mathematician Al-Karaji, and Chinese mathematician Zhu Shijie solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of cubic equations by Fibonacci represented the beginning of the rise in European algebra. Ab? al-? asan bin? Al? al-Qala ?? d? (1412-1486) took "the first step towards the introduction of algebraic symbolism". He also counts? n ² ,? n ³ and uses a sequential approximation method to determine the square root. As the Islamic world declines, the European world is on the rise. And this is where algebra is developed further.

History of modern algebra

FranÃÆ'§ois ViÃÆ'¨te's work on the new algebra at the end of the 16th century is an important step towards modern algebra. In 1637, Renà © ¨ Descartes published La GÃÆ'Ã… © omÃÆ'  © trie , created analytic geometry and introduced modern algebraic notation. Another important event in algebraic development is the general algebraic solution of cubic and quartic equations, developed in the mid-16th century. The idea of ​​a determinant developed by Japanese mathematician Seki K? Wa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on the matrix and determinants of the 18th century. The permutation was studied by Joseph-Louis Lagrange in his 1770 paper Résé © © quérié des quénés devoted to the solution of algebraic equations, where he introduced Lagrange resolvents. Paolo Ruffini was the first to develop a permutation group theory, and like its predecessor, also in the context of solving algebraic equations.

The abstract algebra was developed in the 19th century, which stems from an interest in solving equations, initially focusing on what is now called Galois theory, and on issues of constructiveness. George Peacock is the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovers algebraic relation in his Syllabus of the Logical Proposed System . Josiah Willard Gibbs develops vector algebra in three-dimensional space, and Arthur Cayley develops matrix algebra (this is a non-commutative algebra).

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Mathematical area with algebraic word in their name

Some areas of mathematics that belong to the abstract algebra classification have the word algebra in their name; linear algebra is one example. Others are not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

• Basic algebra, the part of algebra that is usually taught in the math class.
• Abstract algebra, in which algebraic structures such as groups, circles, and planes are defined axiogically and investigated.
• Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.
• Boolean algebra, the algebra branch that abstracts computation with incorrect values ​​ false and is true .
• Commutative algebra, the study of commutative rings.
• Computer algebra, the application of algebraic methods as algorithms and computer programs.
• Homologous algebra, the study of algebraic structures that is fundamental for studying topological space.
• Universal algebra, in which the general properties for all algebraic structures are studied.
• The theory of algebraic numbers, in which the properties of numbers are studied from the algebraic point of view.
• The geometry of algebra, the branch of geometry, in its primitive form determines curves and surfaces as solutions of polynomial equations.
• Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.
• Relational algebra: a set of final relationships that are closed under certain operators.

Many mathematical structures are called algebras :

• Algebra is on the field or more general algebraic on the ring.
  Many classes of algebras on the field or on the ring have a specific name:
  • Associations algebra
  • Non-associative algebra
  • Lie Algebra
  • Hopf algebra
  • C * -algebra
  • Symmetric algebra
  • Exterior algebra
  • Algebra tensor
• In theory of size,
  • Sigma-algebra
  • Algebra in a set
• In category theory
  • F-algebra and F-coalgebra
  • T-algebra
• In logic,
  • Algebraic relations, Boolean algebraic leftover extended with involvement is called otherwise.
  • Boolean algebra, distributed distributing lattice.
  • Heyth Algebra

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Basic algebra

Basic algebra is the most basic form of algebra. It is taught to students who are considered to have no mathematical knowledge outside of the basic principle of arithmetic. In arithmetic, only numbers and their arithmetic operations (such as, -, ÃÆ'â €, ÃÆ' Â ·) occur. In algebra, numbers are often represented by symbols called variables (such as a , n , x , y or < i> z ). This is useful because:

• This allows the general formulation of arithmetic laws (such as a b a b a for all < i> a and b ), and thus is the first step to systematic exploration of the properties of real number systems.
• This allows references to "unknown" numbers, equation formulations and studies on how to solve them. (For example, "Find the number x such that 3 x 1 = 10" or a little farther "Find numbers x like ax b = c "This step leads to the conclusion that it is not the nature of certain numbers that allows us to solve it, but the operations involved.)
• This allows the formulation of functional relationships. (For example, "If you sell x tickets, your profit will be 3 x - 10 dollars, or f ( x ) = 3 x - 10, where f is the function, and x is the number used for the function â €.)
Polynomial

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of a product of a constant and a limited number of variables being raised to the entire power number. For example, x ² 2 x - 3 is a polynomial in a single variable x . A polynomial expression is a rewritable expression as a polynomial, using commutative, association and distribution of additions and multiplications. For example, ( x 3) is a polynomial expression, that, speaking correctly is not a polynomial. A polynomial function is a function determined by a polynomial, or, equivalently, by a polynomial expression. The two previous examples define the same polynomial function.

Two important and related issues in algebra are polynomial factorization, that is, expressing a given polynomial as a product of other polynomials that can not be taken into account any further, and the calculation of the largest common polynomial divider. The above polynomial example can be calculated as ( x - 1) ( x 3). The problem-related class is finding an algebraic expression for the polynomial root in one variable.

Education

It has been argued that basic algebra should be taught to students as young as eleven years, although in recent years it is more common for public studies starting at the eighth grade (? Ã, 13Ã, Â ± yoÃ, Â ±) in the United States.. However, in some US schools, algebra begins in the ninth grade.



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Abstract algebra

Abstract algebra extends the familiar concepts found in basic algebra and numerical arithmetic to a more general concept. Here are listed basic concepts in abstract algebra.

Set : Instead of just considering different numbers of numbers, offer abstract algebra with a more general concept of set : a collection of all objects (called element) is selected by a particular property for the set. All collection of known type numbers is set. Other examples of sets include a set of all two-matrices, a set of all second-degree polynomials ( ax ² bx < ), a set of all two dimension vectors on the plane, and a limited range of cyclical groups, which are modules integers n . Set theory is a branch of logic and is technically not a branch of algebra.

Binary operations : Optional supplements () are abstracted to provide binary operations , * say. The idea of ​​a binary operation is meaningless without a specified set of operations. For the two elements a and b in a set S , a * b is other elements in the set; This condition is called closure. Addition (), subtraction (-), multiplication (ÃÆ'â €), and division (ÃÆ' Â ·) can be a binary operation when defined on different sets, such as the addition and multiplication of matrices, vectors, and polynomials.

Identity element : The zeros and one are abstracted to give an understanding of the identity element for the operation. Zero is the element of identity for addition and the one is the identity element for multiplication. For common binary operators * identity elements e must meet a * e = a and e * a = a , and of course unique, if any. This applies to additions as a 0 = a and 0 a = a and multiplication of a ÃÆ'â € "1 = a and 1 ÃÆ'â € a = a . Not all carrier sets and combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3,...) has no identity element for additions.

Inverted elements : Negative numbers generate the concept of inverse element . In addition, the inverse of a is written - a , and for inverse propagation is written a ^-1 . The general two-sided inversion element a ^-1 satisfies the a a a ^-1 = e and a ^-1 * a = e , where < i> e is the identity element.

Associative : The addition of integers has a property called associativity. That is, the grouping of numbers to be added does not affect the amount. For example: (2 3) 4 = 2 (3 4) . In general, this becomes ( a * b ) * c = a * ( b * c ). This property is shared by most binary operations, but not the reduction or distribution or multiplication of okonion.

Commutative : Addition and multiplication of real numbers is commutative. That is, the order of numbers does not affect results. For example: 2 3 = 3 2. In general, this becomes a * b = b * a . This property does not apply to all binary operations. For example, multiplication of matrices and quaternion multiplication are both non-commutative.

Groups

Combining the above concept gives one of the most important structures in mathematics: a group . Group is a combination of a set of S and one binary * operation, defined in whatever way you choose, but with the following properties:

• The identity element e exists, as it is for every member a of S , e * a and a * e are both identical to a .
• Each element has an inverse: for every member a of S , there are a ^-1 members like that a * a ^-1 and a ^-1 * a are both identical to the identity element.
• This operation is associative: if a , b and c are members S , then ( a * b ) * c is identical to a * ( b * c ).

If the group is also commutative - that is, for every two members a and b of S b is identical to b * a - the group is said to be abelian.

For example, a set of integers under the addition operation is a group. In this group, the identity element is 0 and the opposite of any element a is its state, - a . The requirements of associativity are met, because for each integer a , b and c , (i ) c = a ( b c )

Non-zero rational numbers form groups under multiplication. Here, the element of identity is 1, since 1 = a for each rational number a . Inversion a is 1/i , because a ÃÆ'â € "1/i <1.

The integer under multiplication operations, however, does not form groups. This is because, in general, invers multiplication of integers is not an integer. For example, 4 is an integer, but the inverse multiplication is ¼, which is not an integer.

Group theory is studied in group theory. The main results in this theory are the finite classification of finite groups, mostly published between about 1955 and 1983, separating simple groups down to 30 basic types.

Semigroups, quasigroups, and monoids are structures similar to groups, but more common. They consist of a set and closed binary operations, but do not always meet other conditions. Semigroups have associative binary operations , but may not have an identity element. Monoids are semigroups that do have an identity but may not have an inverse for each element. Quasigroup meets the requirement that each element can be changed to another by a left-hand or a right multiplication; However, binary operations may not be associative.

All groups are monoids, and all monoids are semigroups.

Circles and fields

The group has only one binary operation. To fully explain the behavior of different types of numbers, structures with two operators need to be studied. The most important are the rings, and the fields.

A ring has two binary operations () and (ÃÆ'â € "), with ÃÆ'â €" distributive over. Under the first operator () establishes the abelian group. Under the second operator (ÃÆ'â € ") it is associative, but it does not need to have an identity, or a reverse, so no division is required. The additive identity element () is written as 0 and the additive inversion a is written as - a .

Distributorship separates distributive law for numbers. For integers ( a b ) a < c b ÃÆ'â € "c and c ÃÆ'â €" i> b ) = c c , and ÃÆ'â € "are said to be distributive above.

The integer is an example of a ring. Integer has an additional property that makes it a integral domain .

The field is ring with an additional property that all elements except 0 form a group abelian under ÃÆ'â € ". Multiplicative identities (ÃÆ'â € ") are written as 1 and invers multiplication of a are written as a ^-1 .

Rational numbers, real numbers and complex numbers are all examples of fields.



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See also

• Algebra outline
• Outlines of linear algebra
• Algebraic tiles



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Note




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References

• Boyer, Carl B. (1991), History of Mathematics (second edition), John Wiley & amp; Sons, Inc., ISBNÃ, 0-471-54397-7
• Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
• Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
• George Gheverghese Joseph, The Crest of the Peacock: The Roots of Non-European Mathematics (Penguin Books, 2000).
• John J O'Connor and Edmund F Robertson, Historical Topics: Algebraic Index . In the MacTutor History of Mathematics archive (University of St Andrews, 2005).
• I.N. Herstein: Topics in Algebra . ISBNÃ, 0-471-02371-X
• R.B.J.T. Allenby: Ring, Fields, and Groups . ISBNÃ, 0-340-54440-6
• L. Euler: Algebra Elements , ISBN 978-1-899618-73-6
• Asimov, Isaac (1961). Realm of Algebra . Houghton Mifflin.



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External links

• Khan Academy: Successful conceptual videos and examples
• Khan Academy: The Origins of Algebra, a free online micro college
• Algebrarules.com: Open source resources for learning the basics of Algebra
• Algebra 4000 Years, lecture by Robin Wilson, at Gresham College, October 17, 2007 (available for MP3 and MP4 downloads, as well as text files).
• Pratt, Vaughan. "Algebra". In Zalta, Edward N. Stanford Encyclopedia of Philosophy .

Source of the article : Wikipedia